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Question: In YDSE, bi-chromatic light of wavelength 400nm and 560nm are used. The distance between slits is 0....

In YDSE, bi-chromatic light of wavelength 400nm and 560nm are used. The distance between slits is 0.1mm and distance between the plane of the slits and the screen is 1.0m.The minimum distance between two successive regions of complete darkness is
A. 4mm
B. 5.6mm
C. 14mm
D. 28mm

Explanation

Solution

Hint: the bi-chromatic light is two beams light of two coloured light (as we know the word bi means two)and monochromatic will be single beam and single coloured light. So here it is mentioned that to find the minimum distance between two successive regions it means we have to consider the wavelength that is smaller in numbers.

Complete step by step answer:
Given:
The distance between the slits can be taken asl=0.1 mm×1 m10000 nm=10−4 nml = 0.1\,{\rm{mm}} \times \dfrac{{1\,{\rm{m}}}}{{10000\,{\rm{nm}}}} = {10^{ - 4}}\,{\rm{nm}}
The distance between the plane of the slits and the screen is L=1 mL = 1\,{\rm{m}}
In the both values 400nm and 560 nm, the 400 nm is less, then
the minimum wavelength is taking as λ=400 nm×1 m10000 nm=4×10−7 m\lambda = 400\,{\rm{nm}} \times \dfrac{{1\,{\rm{m}}}}{{10000\,{\rm{nm}}}} = 4 \times {10^{ - 7}}\,{\rm{m}}
Then the minimum distance between two successive regions of complete darkness is taking as β\beta , then
The equation to find the minimum distance between two successive regions is
β=lλL\beta = \dfrac{{l\lambda }}{L}
Substituting the values in the above equation then we will get

\beta = \dfrac{{L\lambda }}{l}\\\ = \dfrac{{1\,{\rm{m}} \times 4 \times {{10}^{ - 7}}\,{\rm{m}}}}{{{{10}^{ - 4}}\,{\rm{m}}}}\\\ = 0.004\,{\rm{m}} \times \dfrac{{1\,{\rm{m}}}}{{1000\,{\rm{mm}}}}\\\ = 4\,{\rm{mm}} \end{array}$$ So we should write the value of $$\beta $$ in terms of mm, then the output will be 4 mm. Therefore, the minimum distance between two successive regions of complete darkness is 0.004 m or 4 mm **So, the correct answer is “Option A”.** **Note:** In the question, it is provided two wavelengths 400 nm and 560 nm as there are two chances but we took only 400 nm as the wavelength to find the required answer because the 400 nm is the minimum value in the both and it is the required input to find. So we should not be stuck with why we took 400 nm whereas there is also 560 nm in the question.